Binomial experiments are very popular for studies because the probability of one possibility or the other can be calculated quickly and accurately. How do you identify a binomial experiment? Can an experiment that is not binomial be easily converted into a binomial experiment?

Look to the end of the lesson for the answer.

### Binomial Experiments

Binomial experiments give rise to **binomial random variables**, which will be the topic of our next couple of lessons. A **binomial experiment** is a very specific type of experiment. In order to be a binomial experiment, there are four qualifications that the experiment must meet:

**There must be a fixed number of trials**. The experiment cannot just be to roll a die until you get a 2, because the number of rolls (trials) is not fixed.**Each trial must be independent of the others.**You cannot have a situation like “If you flip a coin and get heads, flip twice more, and if you get tails, flip three more times”.**Each trial must have a “success” and a “failure”.**Depending on the trial, these may be identified as “yes” and “no” or “0” and “1” or “black” and “white”, etc. However, from a statistics standpoint, the outcome you are studying is generally called the “success” and the other is called “failure”.**The probability of success must be the same for all trials.**The experiment cannot be 10 trials of pulling and keeping a card from a deck to see how many are hearts, because the probability of getting a heart would change each trial. To make this a binomial experiment, you need to replace the card each time.

#### Identifying Binomial Random Variables

1. If a fair coin if flipped 10 times, and \begin{align*}T\end{align*} is the number of tails, is \begin{align*}T\end{align*} a binomial random variable?

Yes, \begin{align*}T\end{align*} is a binomial random variable, and this is a binomial experiment. It meets all four qualifications:

- There is a specific number of trials: 10 flips
- Trials are independent: the outcome of one coin flip does not affect the next flip
- There are only two possible outcomes: a “success” and “failure”. Since we are counting tails, every tails is a “success” and every heads is a “failure”
- The probability is the same for all trials: The probability of getting tails is always 50% if flipping a fair coin

2. If Trina designates \begin{align*}Y\end{align*} to be the number of yellow marbles she gets during nine trials of randomly pulling 1 marble from a bag filled with marbles of various colors and returning it, is \begin{align*}Y\end{align*} a random variable? Is it binomial?

Yes, \begin{align*}Y\end{align*} is a random variable, since it is the random numerical result of a limited number of independent trials of an experiment. It is also binomial, since each of the limited trials is independent, has a success/failure (yellow/not yellow), and has the same probability of success.

#### Finding Unknown Values

If \begin{align*}N\end{align*} is the number of nines you get when rolling two standard dice three times:

- Is \begin{align*}N\end{align*} a binomial random variable?
- What are the possible values of \begin{align*}N\end{align*} ?
- Create a histogram or pie chart showing the probability distribution of \begin{align*}N\end{align*}.

a. Is \begin{align*}N\end{align*} a binomial random variable?

\begin{align*}N\end{align*} is a binomial random variable, because it is the result of a specific and limited number of independent trials of a random process and each outcome is either nine or not nine.

b. What are the possible values of \begin{align*}N\end{align*}?

Since you could only roll a total of 9 once each trial, \begin{align*}N\end{align*} could be 0, 1, 2, or 3.

c. Create a histogram or pie chart showing the probability distribution of \begin{align*}N\end{align*}.

The probabilities of each of the possible values of \begin{align*}N\end{align*} would be:

(see the lesson: Understanding Discrete Random Variables, Example C, for the calculations)

- \begin{align*}N=0: \frac{512}{729}\end{align*}
- \begin{align*}N=1: \frac{192}{729}\end{align*}
- \begin{align*}N=2: \frac{24}{729}\end{align*}
- \begin{align*}N=3: \frac{1}{729}\end{align*}

A pie chart would look like this: (note that total probability = \begin{align*}\frac{512}{729} + \frac{192}{729} + \frac{24}{729} + \frac{1} {729} = \frac{729}{729}\end{align*}).

#### Earlier Problem Revisited

*Binomial experiments are very popular for studies because the probability of one possibility or the other can be calculated quickly and accurately. How do you identify a binomial experiment? Can an experiment that is not binomial be easily converted into a binomial experiment?*

A binomial experiment must consist of a limited number of independent trials, where each trial outcome is either a success or a failure, and each trial has the same probability of success as all other trials.

A non-binomial experiment can often be viewed as binomial by carefully stating the outcome of each trial in a binomial format. For example, a non-binomial experiment might be “Count the number of heads and tails resulting from 8 flips of a fair coin”. Viewed as a binomial experiment, the same results could be collected from “How many tails do you get by flipping a fair coin 8 times”? You could then subtract the result from 8 to get the number of “not tails”, e.g. “heads”.

### Examples

#### Example 1

Mariska spins a spinner 40 times, recording the number of 4’s she gets. Is this a binomial experiment?

Yes, this is a binomial experiment because Mariska is conducting a limited number of independent random "4" or "not 4" trials, and the probability of spinnig a "4" does not change.

#### Example 2

Heidi has a bag containing 4 blue, 3 green, 5 red, and 7 yellow marbles. She defines a trial as pulling a marble, recording the color, and replacing it. She records the number of trials it takes to pull a green marble. Is this a binomial experiment?

No, Heidi is not conducting a binomial experiment because the number of trials is not specified, she just keeps pulling until she gets a green.

#### Example 3

Evan notes that 24% of online game players he polled are between 30 and 39 years old. Evan decides to create a team of players from that age range by randomly choosing names from among those he polled, keeping each one he chooses that is in his/her 30’s. If he chooses a name only 10 times, no matter the number of players he gets, is this a binomial experiment?

No, Evan is not conducting a binomial experiment because the probability that a random player will be between 30 and 39 changes each tim ehe keeps one for his team.

### Review

For questions 1-12, state that a particular experiment is or why it is not binomial:

- A spinner has a 35% probability of landing on blue. Let \begin{align*}B\end{align*} be the number of blues spun in 5 spins.
- A bag contains 6 blue, 4 green, and 3 red candies. Let \begin{align*}G\end{align*} be the number of green candies you pull out and eat in 5 trials.
- One trial of an experiment consists of pulling a random card from a standard deck, noting it, and replacing it, you conduct 12 trials.
- One trial consists of pulling two cards from a standard deck, noting them, and replacing them. Let \begin{align*}T\end{align*} be the number of trials until you pull two face cards at the same time.
- A 20-sided die is rolled ten times, and \begin{align*}S\end{align*} is the number of sevens rolled.
- Assume that 15% of word game players create at least 12 words out of 50 that have more than 5 letters, and you let \begin{align*}W\end{align*} be the number of letters in words from 20 trials of 1 game each.
- A die is rolled 20 times. What is the probability of rolling a 1 exactly 5 times?
- You plan on choosing students (with replacement) from a population of 28, 17 of which are Juniors. You want to know how many will have to be picked before getting a Junior.
- A new reality show is so popular that an estimated 47% of households watch it every week. You choose 20 households at random. Let \begin{align*}X\end{align*} be the number of households watching the show.
- \begin{align*}H\end{align*} is the number of heads tallied over ten flips of a fair coin.
- \begin{align*}F\end{align*} is the number of 5’s you roll before rolling a 6, on a standard die.
- \begin{align*}O\end{align*} is the number of 1’s you roll in fifteen rolls of a standard die.

### Review (Answers)

To view the Review answers, open this PDF file and look for section 7.6.